Prove the exact joint T29 bin-count spectrum (no new κ)

research/t29-joint-bin-spectrum/README.md

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+# Exact joint T29 bin-count spectrum
+
+This contribution was produced by Nick Petalas. Apache-2.0.
+
+Claim type: scoped constructive proof with exact finite corroboration. In the
+historical fixed U8 whole-helper packing model at T=29, let x count
+capacity-16 `{9,7}` bins and y count capacity-8 `{4,4}` bins. The exact joint
+projection is the full integer rectangle
+
+```text
+4824 <= x <= 12992,       0 <= y <= 8286.
+```
+
+Every pair in this rectangle occurs. There is no additional coupling or
+parity restriction between these two counts. This answers the explicit Open
+next question in woklerx's [PR1844](https://github.com/CrocSwap/integer-mult-bounds/pull/1844).
+The new result is the joint projection and its inventory-preserving exchange.
+The predecessor's one-dimensional x range and starting construction are cited
+premises, not new results or audit targets. No kappa improvement is claimed.
+
+## Model and cited premises
+
+The capacity quotas at `(8,16,32,48)` are `(19488,12992,12992,12992)`.
+The helper demands at widths `(4,5,7,8,9)` are
+`(20880,8352,14094,31842,56376)`. The required remainder profile is
+`{0:49770,1:4,2:1,48:8689}`. Helpers remain whole. No chronological,
+geometric, chart or physical owner constraint is imposed.
+
+Hanhong Zhao's [PR1160 proof](../t29-capacity-audit/PROOF.md), Lemmas 1-2,
+shows that every realization uses tight patterns. Its four remainder-one
+bins each contain one four, its remainder-two bin contains at least one
+four, and each of its 4,303 full capacity-48 bins contains at least one four.
+It also proves the necessary x range. We use these statements as published
+premises, without rerunning their pattern census.
+
+Woklerx's [PR1844 construction](../t29-capacity-projection-spectrum/README.md)
+gives a realization for every integer x in that range with y=8286. Write
+`x=4824+2t+e`, where e is zero or one. Its groups outside capacity eight
+that contain eights are exactly:
+
+| Capacity | Pattern | Number of bins | Eights per bin |
+|---:|---|---:|---:|
+| 16 | `{8,8}` | `8168-2t-e` | 2 |
+| 32 | `{9,9,8,4}` | 1 | 1 |
+| 32 | `{9,8,8,7}` | e | 2 |
+| 32 | `{8,8,8,8}` | t | 4 |
+| 48 | `{9,9,9,9,8,4}` | 4303 | 1 |
+
+All other groups are retained unchanged. Exact proof and construction bytes
+are pinned as historical provenance in [SOURCES.json](SOURCES.json).
+
+## Proof of the joint range
+
+Necessity of the x interval is PR1160's result. Nonnegative counts give
+`y>=0`. The forced bins cited above consume at least `4+1+4303=4308`
+fours outside capacity eight. The y distinguished bins consume another 2y,
+so `2y+4308<=20880`, giving `y<=8286`. This is a new necessary bound on y.
+
+For sufficiency, start with PR1844's realization for the desired x. The table
+contains exactly
+
+```text
+2(8168-2t-e) + 1 + 2e + 4t + 4303 = 20640
+```
+
+eights outside capacity eight, independently of x. Replacing an eight in any
+one bin by two fours preserves its total width, capacity tag and remainder.
+It also preserves its signed price since an eight and two fours both cost
+-2 in the inherited price system. Any subset of these 20,640 occurrences
+can be replaced, including several occurrences within a bin.
+
+Choose any desired integer `0<=y<=8286` and put `k=8286-y`. In exactly k
+of the starting capacity-8 `{4,4}` bins, replace the two fours by one eight.
+In exactly k of the listed external eight occurrences, make the reverse
+replacement. There are enough occurrences because `k<=8286<20640`.
+The two changes cancel in the four and eight inventories. They leave all
+other helper demands, all bin quotas and the entire remainder profile fixed.
+
+The external move cannot change the number of capacity-16 `{9,7}` bins:
+those bins contain no eights, and replacing eights creates only fours.
+The capacity-eight move reduces y by exactly k. Thus the result has the
+prescribed pair `(x,y)`. This proves sufficiency for every integer pair,
+without interpolation or an integral-relaxation assumption.
+
+The verifier implements the construction using counts of equal patterns.
+For each group with n bins and s eights per bin, spending a replacements
+converts `floor(a/s)` complete bins, at most one partial bin, and leaves the
+rest unchanged. This realizes every integer budget a between zero and ns.
+No individual-helper splitting is used.
+
+## Verification and reproduction
+
+From the repository root, with CPython 3.12 and the standard library:
+
+```sh
+/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -W error -B research/t29-joint-bin-spectrum/verify.py
+/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -W error -O -B research/t29-joint-bin-spectrum/verify.py
+/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/t29-joint-bin-spectrum
+/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/t29-joint-bin-spectrum --check-source-history
+```
+
+The direct commands emit `PASS_T29_JOINT_BIN_SPECTRUM` in a generated JSON
+receipt bound to every package file's SHA256. They write no files and import
+no predecessor executable. The finite checks construct new joint witnesses
+and check every capacity, every helper width, every remainder and both named
+projections. They test the external pool at every x, every y on four x slices,
+and six boundary/interior y values at each remaining x. Local exchange checks
+cover every convertible starting shape. Malformed coordinates, missing
+compensation, an extra bin and a false projection must reject under normal
+and optimized Python. Universal coverage follows from the written exchange
+proof; the sampled interior grid is corroboration, not exhaustive pair coverage.
+
+Local normal and optimized runs each pass 82,138 joint constructions, 8,169
+pool identities, 15 local exchange checks and 11 rejection controls. The two
+registered commands took 1.65 and 1.65 seconds on CPython 3.12.15/macOS arm64.
+These are check durations, not performance improvement measurements.
+
+The selected source-binding orientation also passes with
+`scripts/research_selected_baseline.py`. No selected operator replay,
+predecessor census, Lean or Docker build is run. Independent human and formal
+proof-kernel review are unclaimed.
+
+## Scope and implementation bounds
+
+The selected-packing card changes to this historical arithmetic side question.
+The joint range describes assembly freedom within a known sharp remainder
+profile. It supplies no additional recursive saving and makes no claim about
+the changed selected #4470 supplier. Physical matching, owner geometry,
+chronology, charts, storage, precision and a complete caller remain unpriced.
+No further construction expansion follows from this result.
+
+The verifier has no recursion, subprocesses, network access or generated code.
+It retains one construction at a time, with at most 36 pattern groups and
+12 widths per group. Its outer loops range over 8,169 fixed x coordinates and
+at most 8,287 y coordinates. Each replacement loop has at most four steps.
+Functions remain below 60 physical lines; explicit exceptions retain all
+contracts under optimization. Warning-strict execution supplies diagnostics.
+AST syntax and function-size inspection pass. No standalone linter or static
+type checker is installed in the execution environment.
+CPython's dynamic objects and integer arithmetic are a managed-runtime
+deviation from Power of Ten allocation constraints, bounded here by these
+fixed domains. No speed or memory benchmark claim is made.
+
+## Open next questions
+
+1. What is the joint projection after also fixing the number of full
+   capacity-48 `{9,9,9,9,8,4}` bins? This would distinguish the remaining
+   exchange freedom from a genuinely coupled constraint. Start by retaining
+   that coordinate in the same four/eight ledger. The present move changes it,
+   so the rectangle theorem does not settle the three-coordinate problem.
+2. Can any realization in this rectangle satisfy one literal historical owner
+   and chronological constraint? Start with one named owner fixture and its
+   legal windows. These constraints and their physical prices are absent here.
+
+## Credit
+
+Woklerx supplies PR1844's affine family and the explicit joint-projection
+question. Hanhong Zhao supplies PR1160's capacity model, tight-pattern theorem
+and capacity-tagged realizations. Chafik Boukhalfa supplies the PR795/PR890
+prices and family; Jonathan Capalbo supplies PR942's all-replica relaxation;
+woklerx supplies PR992's sharp T29 profile; Hao Yan supplies the PR858 owner
+lineage. The new joint-range proof and verifier were produced by Nick Petalas.
+All predecessor files and credit lines remain unchanged.

research/t29-joint-bin-spectrum/SOURCES.json

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+{
+  "schema_version": 1,
+  "base_commit": "81e9bd69e4a5b00ba9c713a1620e36d576d5522e",
+  "files": [
+    {
+      "path": "certificates/selected-result.json",
+      "usage": "provenance",
+      "git_blob": "c35482cd9035843b15a634cc8276a7e1e40e8395",
+      "sha256": "6fe0a099b89aa2aed796a13c3a21dafc63f3388ebf397464e44e0451893b9375"
+    },
+    {
+      "path": "docs/research/agent-task-catalog.json",
+      "usage": "provenance",
+      "git_blob": "f126dd2e2dad2b669e75d1d0d0bd05c600725602",
+      "sha256": "f7a48bfa2659b88378282a8eee7d9fe52cdf505594124c4d1686b58510fffd72"
+    },
+    {
+      "path": "research/t29-capacity-audit/PROOF.md",
+      "usage": "provenance",
+      "git_blob": "9013468c524b9c740de01669d2a246abd8bc763e",
+      "sha256": "9fc548c21d2d7ee8b23dcb67d9001b8bc604f1e2a7bea32f3303b12e5053a02e"
+    },
+    {
+      "path": "research/t29-capacity-projection-spectrum/README.md",
+      "usage": "provenance",
+      "git_blob": "9e888d164ec097d1b618bfc5329e4f2d3ad6b625",
+      "sha256": "32ed9e6379d5fe905b9e65f5be3044e3a1f6d2b62d264e3852842faf716528e2"
+    }
+  ]
+}

research/t29-joint-bin-spectrum/TASK.json

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+{
+  "schema_version": 1,
+  "task_id": "selected-packing",
+  "priority_revision": 7,
+  "base_commit": "81e9bd69e4a5b00ba9c713a1620e36d576d5522e",
+  "original_card": {
+    "id": "selected-packing",
+    "question": "Does a named residual packing task have meaningful uncovered headroom after its exact fixed-inventory ceiling?",
+    "starting_points": [
+      "docs/research/research-family-status.md",
+      "certificates/selected-result.json",
+      "research/staged-fourth-mixture-ceiling/README.md",
+      "research/staged544-mixed-fourth-entry/README.md"
+    ],
+    "methods": [
+      "Check a fixed-inventory ceiling and source equivalence before any explicitly bounded residual search.",
+      "Price a genuinely changed word/interface under a newly applicable model."
+    ],
+    "acceptance_gate": "Check live overlap and exact model/source binding first. Distinguish a chronological construction from a pooled optimum. State the applicable fixed-inventory ceiling and remaining headroom; changed words or interfaces require a new fully charged model. Do not launch a residual search without an explicit scope and time cap.",
+    "first_step": {
+      "command": [
+        "python3",
+        "-B",
+        "scripts/research_baseline.py"
+      ],
+      "expected": "PASS_RESEARCH_BASELINE_BINDINGS_ONLY",
+      "inspect": "Read #4091 fixed-word pooled ceiling and #4088 later 1090-call literal fill. Bind the unchanged local invoices/capacities/stock before transferring the ceiling; its earlier 1080-call residual-bin recipe is historical.",
+      "scope": "Source-binding orientation only; no scientific replay, theorem acceptance or supplier admission. Budget any heavy replay separately."
+    },
+    "first_experiment": "Default to a bounded headroom/source-equivalence check. Only start residual search with an explicit unclaimed domain, time cap and reason the tiny remaining gain matters. Otherwise hand off or select a changed-word mechanism. A changed word or interface needs a fresh cost model, not the old ceiling. Stop at the fixed-inventory cap; do not extend by new random seeds.",
+    "bottleneck_link": "An actual paid-profile gain can improve the frontier up to the other supplier cap.",
+    "allowed_variations": [
+      "One bounded residual packing instance or exact relaxation.",
+      "If the coverage law, template or guard changes, derive an applicable fully charged replacement bound instead of applying the incumbent ceiling outside its scope."
+    ],
+    "scope_change_triggers": [
+      "Rerunning an existing search with only a different seed as a separate mathematical contribution."
+    ],
+    "overlap_terms": [
+      "T16",
+      "owner exchange",
+      "packing",
+      "4091",
+      "4088",
+      "staged-fourth-mixture-ceiling",
+      "staged544-mixed-fourth-entry"
+    ],
+    "existing_checks": [
+      {
+        "scope": "PR #4091: pooled optimum at one-third staging for pinned fixed words; not a chronological fill.",
+        "evidence": "research/staged-fourth-mixture-ceiling/README.md",
+        "remaining": "Independent scope/source check; any residual optimization is explicitly bounded and optional.",
+        "status": "author-checked",
+        "source_commit": "a29bdd8f56cd18b805917775cdec09ff935eddc4",
+        "selected_record_sha256": "bb2eaa58647b353f88d0aec2eafb9d30df1be53589231b23e12ceef3e4bad129"
+      },
+      {
+        "scope": "PR #4088: later 1090-call literal fill improves the earlier 1080-call witness.",
+        "evidence": "research/staged544-mixed-fourth-entry/README.md",
+        "remaining": "Complete admission remains separate; old residual-bin instructions must not be copied blindly.",
+        "status": "author-checked",
+        "source_commit": "72d7efc181edb21806a878f174847c96dd410ab0",
+        "selected_record_sha256": "bb2eaa58647b353f88d0aec2eafb9d30df1be53589231b23e12ceef3e4bad129"
+      }
+    ]
+  },
+  "scope_status": "changed",
+  "actual_scope": "Exact joint projection of capacity-16 {9,7} and capacity-8 {4,4} counts in the historical fixed U8 T=29 packing model.",
+  "scope_change_reason": "Takes the explicit open question in woklerx PR1844 rather than auditing fixed selected invoices or searching a residual packing. The result concerns arithmetic assembly freedom and has no selected-profile gain.",
+  "bottleneck_link": "Side result: identifies exactly which two inventory coordinates remain independent below physical scheduling. It does not improve the selected construction or its recursive bill.",
+  "budget_minutes": 60,
+  "stop_rule": "Stop at the first verified joint-range theorem. Leave extra coordinates and physical owner constraints open.",
+  "outcome": "completed",
+  "evidence": "README proof plus standalone exact new joint constructions, complete inventory/remainder/projection invoices and rejecting controls; predecessor tight-pattern theorem and affine family are cited premises.",
+  "author": "Nick Petalas",
+  "ai_assistance": "Independent human and formal proof-kernel review are unclaimed.",
+  "open_questions": [
+    {
+      "question": "Determine the joint projection with the full capacity-48 {9,9,9,9,8,4} count also fixed.",
+      "why_it_matters": "Distinguishes remaining exchange freedom from a genuinely coupled constraint.",
+      "first_step": "Retain this third coordinate in the same four/eight exchange ledger.",
+      "blocked_by": "The present exchange can change that count; no necessity or construction with it fixed is supplied."
+    },
+    {
+      "question": "Realize one pair under a literal historical owner and chronological constraint.",
+      "why_it_matters": "The arithmetic rectangle alone cannot establish physical feasibility.",
+      "first_step": "Name one owner fixture and encode its legal windows before pricing it.",
+      "blocked_by": "Geometry, charts, windows, restoration and physical costs are absent from this model."
+    }
+  ],
+  "coordination": {
+    "claim": "https://github.com/CrocSwap/integer-mult-bounds/discussions/1166#discussioncomment-19142736",
+    "upstream_check_utc": "2026-10-23T09:34:00Z",
+    "predecessors": [
+      {
+        "pr": 1844,
+        "author": "woklerx",
+        "scope": "Affine x family and explicit joint-projection question"
+      },
+      {
+        "pr": 1160,
+        "author": "Hanhong Zhao",
+        "scope": "Tight-pattern theorem and necessary x interval"
+      }
+    ]
+  },
+  "claim_type": "scoped constructive proof with finite corroboration"
+}

research/t29-joint-bin-spectrum/VERIFY.json

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+{
+  "schema_version": 1,
+  "status": "ready",
+  "python": "3.12",
+  "requirements": [],
+  "inputs": [],
+  "commands": [
+    {
+      "argv": [
+        "{python}",
+        "-W",
+        "error",
+        "-B",
+        "{package}/verify.py"
+      ],
+      "timeout_seconds": 120,
+      "checks": "New joint constructions, full capacity/helper/remainder/projection invoices, all-x pool identities, local exchange invariants and rejecting controls."
+    },
+    {
+      "argv": [
+        "{python}",
+        "-W",
+        "error",
+        "-O",
+        "-B",
+        "{package}/verify.py"
+      ],
+      "timeout_seconds": 120,
+      "checks": "New joint constructions, full capacity/helper/remainder/projection invoices, all-x pool identities, local exchange invariants and rejecting controls."
+    }
+  ]
+}